The statistical properties of fracture strength of brittle and quasibrittle materials are often described in terms of the Weibull distribution. However, the weakest-link hypothesis, commonly used to justify it, is expected to fail when fracture occurs after significant damage accumulation. Here we show that this implies that the Weibull distribution is unstable in a renormalization-group sense for a large class of quasibrittle materials. Our theoretical arguments are supported by numerical simulations of disordered fuse networks. We also find that for brittle materials such as ceramics, the common assumption that the strength distribution can be derived from the distribution of preexisting microcracks by using Griffith's criteria is invalid. We attribute this discrepancy to crack bridging. Our findings raise questions about the applicability of Weibull statistics to most practical cases. © 2014 American Physical Society.